How to move around the gradient spline
You can key-frame the location field(s) in the Mapping node. 

At the moment you are mapping in Camera space, (i.e. the space of the whole rendered screen in XY, and Z in and out of it.) Maybe one of the spaces associated with your object would suit your purposes better, it depends what you want to do. 

Select 'Texture' in the Mapping node, and right-click in any of the  Location fields to set keyframes, and  animate the position of the texture in X/Y/Z.

Frame 5:



Frame 40: